Habitable Zones with Stable Orbits for Planets around

Mon. Not. R. Astron. Soc. 000, 1–?? (—-) Printed 23 May 2014 (MN LATEX style file v2.2)

Habitable Zones with Stable Orbits for Planets around BinarySystems

Luisa G. Jaime1?, Luis Aguilar2, and Barbara Pichardo11Instituto de Astronomıa, Universidad Nacional Autonoma de Mexico, Apdo. postal 70-264, Ciudad Universitaria, Mexico2Instituto de Astronomıa, Universidad Nacional Autonoma de Mexico, Apdo. postal 877, 22800 Ensenada, Mexico

Accepted. Received ; in original form

ABSTRACT

A general formulation to compute habitable zones around binary stars is presented. Ahabitable zone in this context must satisfy two separate conditions: a radiative one and one ofdynamical stability. For the case of single stars, the usual concept of circumstellar habitablezone is based on the radiative condition only, as the dynamical stability condition is taken forgranted (assuming minimal perturbation from other planets). For the radiative condition, weextend the simple formulation of the circumstellar habitable zone for single stars, to the caseof eccentric stellar binary systems, where two sources of luminosity at different orbital phasescontribute to the irradiance of their planetary circumstellar and circumbinary regions. Ourapproach considers binaries with eccentric orbits and guarantees that orbits in the computedhabitable zone remain within it at all orbital phases. For the dynamical stability condition,we use the approach of invariant loops developed by Pichardo et al. (2005) to find regionsof stable, non-intersecting orbits, which is a robust method to find stable regions in binarystars, as it is based in the existence of integrals of motion. We apply the combined criteriato calculate habitable zones for 64 binary stars in the solar neighborhood with known orbitalparameters, including some with discovered planets. Formulae and interpolating tables areprovided, so the reader can compute the boundaries of the habitable zones for an arbitrarybinary system, using the stellar flux limits they prefer. Together with the formulae providedfor stable zones, these allow the computation of both regions of stability and habitabilityaround any binary stellar system. We found 56% of the cases we consider can satisfy bothrestrictions, this is a very important constriction to binary systems. Nevertheless, we concludethat these systems where a dynamical and radiative safe zone exists, must be considered strongcandidates in the search for habitable planets.

Key words: binaries: general, planets, habitability

1 INTRODUCTION

The discovery of hundreds of extrasolar planets has hurled the topicof planetary studies into centerstage: planet formation, planetarydynamics, planetary geology, etc.; among these studies, the ques-tion of planet habitability is of great interest, even if no hard databeyond Earth exists, yet. The quest for life on other planets startedlong ago when in the 60’s, Frank Drake used the 85-foot radiustelescope in West Virginia, hoping to detect an extraterrestrial sig-nal (Drake 1961). However, extraterrestrial life, if it indeed ex-ists in the neighborhood of the Sun, is most likely of a basic type(e.g. microbial). Although a new emerging view is that planets verydifferent to Earth may have the right conditions for life, which in-creases future chances of discovering an inhabited world (Seager2013), it is a good first step along this endeavor to find out if condi-

? E-mail: [email protected] (LGJ)

tions propitious to Earth-like planetary life (the only type we knowfor sure so far) exists.

Even in the case of Earth, life exists in many diverse environ-ments, from scorching deserts to the eternal darkness of the oceandepths, and even deep within Earth’s crust. Confronted with thisbewildering diversity of environments, we must look for the mostbasic ingredients, like liquid water. On the other hand, our obser-vational limitations to detect the most important habitability indi-cator: water vapor on terrestrial-like exoplanets, reduces our possi-bilities of finding habitable planets. Up to now, we have been ableto characterize habitable zones mainly around single stars, conse-quently, the quest for habitable worlds has been mostly limited tothis type of stars. But given the fact that most stars appear to be partof binary and multiple systems, we must extend the habitable-zoneconcept to those cases.

The condition for the existence of liquid water on the surfaceof a planet (Bains 2004), is the usual defining condition for what

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is now called the circumstellar habitable zone (CHZ). Hart (1979)studied the limits of the CHZ in late type stars and concluded that Kdwarfs have a narrow CHZ and later dwarfs have none at all. How-ever, other studies that include among other things, atmosphericradiative transfer modeling, have come with a more optimistic and,rather complicated outlook (e.g. Doyle et al. (1998)). In this case,habitability seems to be a very planet-specific matter (Seager 2013).For example, the habitable zone calculations defined for a dry rockyplanet, with a minimum inner edge of about 0.5 AU, for a solar-like host star (Zsom et al. (2013)), out to 10 AU for a planet withan H2 atmosphere and no interior energy, around a solar-like hoststar (Pierrehumbert & Gaidos 2011), and even possibly out to free-floating habitable planets, with no host star, for planets with thickH2 atmospheres (Stevenson 1999).

For the case of planets in binary stellar systems, the usual ra-diative condition used to define habitable zones around single starsmust be extended to the case of two sources of illumination whoserelative positions change in time. Additionally, a condition of dy-namical stability must be added, as the more complex and timevarying potential turns unstable whole swaths of phase space. Fora planet to be a possible adobe of life, both conditions must be sat-isfied, and for a long time, enough for life to develop.

In the present work we will use Kopparapu et al. (2013) defi-nition of habitable zone around single stars and extend it to the caseof binary stars. As such, this is a simple definition of the habitablezone based on stellar flux limits. Some of the more complicated(and realistic) effects that pertain to particular situations, can beincorporated into this simple criterion by the use of “corrective fac-tors” applied to the stellar fluxes that define the limits of the hab-itable zones. An example of this is provided by e.g. Kaltenegger& Haghighipour (2013), who introduce a factor called “spectrallyweighted flux”.

Studies like the previous ones, have been applied tradition-ally to single stars or brown dwarfs, however, most low-mass main-sequence stars are members of binary or multiple systems (Duquen-noy & Mayor (1991); Fischer & Marcy (1992)), and in particular inthe Solar Neighborhood, the fraction goes up to 50% (Abt (1983)Raghavan et al. (2010)). This suggests that binary formation is atleast as probable as single star formation processes (Mathieu 1994).Additionally, several types of planets have been discovered in cir-cumstellar orbits, even in close binary systems, where the effectof the stellar companion might be of great importance (Dumusqueet al. (2012); Chauvin et al. (2011); Muterspaugh et al. (2010);Correia et al. (2005); Zucker et al. (2004); Hatzes et al. (2003);Queloz et al. (2000), and also in circumbinary discs (Doyle et al.(2011); Welsh et al. (2012); Orosz et al. (2012a); Orosz et al.

(2012b); Schwamb et al. (2013)). For a review on the subject seeHaghighipour (2010); and Kaltenegger & Haghighipour (2013).

We provide in this work a general theoretical formulation tocalculate the equivalent of the Kopparapu et al. (2013) habitablezone for single stars, but for binary systems (see section 2). Thisformulation is expressed in formulae and interpolating tables thatthe reader can apply to any specific case. Our work also consid-ers binary systems with no restriction on the eccentricity of thebinary orbits and guarantees that orbits within the computed habit-able zones remain so, at all binary orbital phases. We only considerapproximately circular habitable zones. For the dynamical stabilitycondition, we use the work of Pichardo et al. (2005) and Pichardoet al. (2009), who found the extent of stable, non-intersecting orbitsaround binary systems, based on the existence of sturdy structuresin the extended phase-space of the system (the so called “invariantloops”). We have then applied both conditions to a sample of main

sequence binary systems in the solar neighborhood with known or-bital parameters and present our results. These constitute regionswhere we think it is most likely to find planets suitable for life inthese binary systems.

Several other interesting papers have been submitted on thissubject recently, for circular binary systems (Cuntz (2013)), andfor the general case of elliptical binary systems (Eggl et al.(2012); Eggl et al. (2013); Haghighipour (2010); Haghighipour &Kaltenegger (2013); Kaltenegger & Haghighipour (2013); Muller& Haghighipour (2014); Kane & Hinkel (2013)). However allthese formulations take as the stability criterion that of Holman& Wiegert (1999). The empirical approximation of Holman &Wiegert to the stability problem in binaries, was an excellent andfundamental first approximation to the solution. Their work wasbased on a detailed trial and error technique to find the most stable(long-term) orbits in disks around binaries. The authors defined asstable orbits mostly those that would keep in their orbits for about104 binary periods (in general not enough to ensure life emergencyand development though). On the other hand, and as the authorspoint out explicitly in their work, their formulation for stability re-gions breaks toward higher eccentricity binaries due to the rapidlyincreasing difficulty to recover stable orbits from this methodology.

In this paper, we rely on the invariant loops theory, which bydefinition are invariant structures in the extended phase-space ofthe system. The existence of invariant loops guarantees the stabil-ity of orbits for all times and not just during the integration span, aslong as the binary parameters (stellar masses, semi-major axis andorbital eccentricity) do not change. Their stability is given by theconstants of motion that support them. The orbit integration is onlyused to identify these structures in extended phase-space (for de-tails see Pichardo et al. (2005); Maciejewski & Sparke (2000); Ma-ciejewski & Sparke (1997); Arnold (1984); Lichtenberg & Lieber-man (1992)). Because of its nature, this formulation has no restric-tions in eccentricity or mass ratio. With this criterion we search forthe intersection between the stable regions for planets and a short,straightforward, and also general formulation for habitability. Ad-ditionally, it is relevant to mention here, that the use of the invariantloops criterion shows, for example in the circumbinary disks case,important differences in the position of stable regions (Pichardo etal. 2009) with respect to previous work, that should be taken intoaccount in habitability calculations. We explain this in detail alongthis paper.

In the second part of this work we also provide a sample ofbinary systems of the solar neighborhood (the whole sample of bi-nary stars, in the main sequence, with all orbital parameters knownat the present time in literature), with their habitable zones com-puted using the approach in this work for habitability and planetaryorbital stability.

This paper is organized as follows. In section 2 the radiativecondition for habitability is extended from the usual circumstellarcase, and a detailed description of the formulation used to calculatethe radiative safe zones is provided. In section 3 we present thedynamical stability condition for habitability used in this work. 4,the combined conditions are used to compute habitability zones forparticular cases and the construction of a table of binaries, withknown orbital parameters, for stars in the main sequence. Finally,our conclusions are given in Section 5.

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Habitable Zones with Stable Orbits for Planets around Binary Systems 3

2 RADIATIVE CONDITION FOR HABITABLE ZONESIN BINARY STARS

A common definition of the CHZ uses limits in the radiative stellarflux at the planet (e.g. Kopparapu et al. (2013)):

Io 6 I(r) 6 Ii, (1)

where the local flux is given by the stellar luminosity divided bydistance squared: I = L/r2, and Io, Ii, define the outer and innerboundaries of the CHZ, respectively.

The CHZ thus defined is a thick spherical shell that surroundsthe star, whose thickness and size depend on the star’s luminosity.However, an important fraction of stars in the solar neighborhoodare part of binary systems. The fact that planets have already beendiscovered within binary systems, makes it necessary to extend thesimple definition of the CHZ to the stellar binary case.

In this section we extend the simple CHZ condition for singlestars given by the previous equation, to the case of stellar binarysystems. In this case we have two sources of luminosity at positionsthat change with the binary orbital phase. It may be thought that inthis case it is simply a matter of applying the equation twice, oncefor each star. However, this naive approach is not correct, sincethere may be regions where, although within the individual CHZfor each star, the combined irradiance of both stars may push theregion out of the combined habitable zone. Additionally, it is funda-mental to add the condition of orbital stability, as both, the correctirradiance and orbital stable regions should have a non-empty in-tersection during the entire binary orbital phase, for planets to beable to exist within a proper binary habitable zone. In Figure 1,we present a schematic figure that shows the combined concept toconstruct habitable zones in a binary star: the radiative safe zonein gray circles (upper half of the diagram), and the stable regionsfor orbits to settle down (lower half of the diagram). Notice howthe demand that both conditions (radiative and dynamical) are met,severely restricts the resulting BHZ.

Since what matters is the total combined irradiance at a givenpoint, we should add the individual stellar fluxes in the CHZ condi-tion given by equation (1), to arrive at the condition for the radiativesafe zone. The total stellar flux is given by:

I(x, y) = LT

[(1− λs)

(x− rp)2 + y2+

λs

(x− rs)2 + y2

], (2)

here LT is the total binary luminosity and λs is the fractional con-tribution of the secondary star to it. x and y are Cartesian coordi-nates in the binary orbital plane and rp and rs are the primary andsecondary star distances to their barycenter. The x-axis containsboth stars.

To illustrate the concept of binary habitable zone, we showin fig. 1 a schematic diagram that presents the radiative condition(upper half) and dynamical condition (lower half). The stars are theblack dots on the x-axis, whose size is proportional to the assumedluminosity (upper half), or mass (lower half). In the upper half,isopleths of constant combined stellar flux are shown, with thosecorresponding to the boundaries of the raditaive safe zone shownwith thick black lines. The raditaive safe zone is shaded in gray.Likewise, in the lower half of the diagram, equipotentials are shownwith the circumbinary and two circumstellar dynamical safe zonesin gray. Notice that the saddle point between the stars for isoplethsand equipotentials does not coincide, as the former is set by therelative luminosities, whereas the latter by their mass ratio.

We define our binary habitable zones (BHZ) as the annular

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Figure 1. Illustrative diagram for the concept of binary habitable zone. Thestars are on the x-axis (black dots whose size is proportional to their lu-minosity —upper half—, or mass —lower half—-). The upper part of thediagram shows the flux isophotes (equation 2) with the boundaries of theradiative safe zone shown with thick black lines and the zone itself shadedin gray. The lower part shows the equipotentials, with the circumbinary andcircumstellar zones of dynamical stability sown in gray. Our definition ofbinary habitable zones (BHZ) is shown as the annular green region, whereboth conditions are met. In this case there is a circumbinary and a circum-secondary habitable zone.

regions (shown in green) where both conditions are met. For thecircumbinary habitable zone, it is an annular region centered atthe barycenter of the system. For the circumstellar habitable zones,they are annular regions centered in the corresponding star. Noticethat in this particular example there is a circumbinary and a circum-secondary habitable zones, but not a circumprimary. In this sectionwe will tackle the radiative condition alone, leaving the conditionof dynamical stability for the next section and the combined effectfor section 4.

The edges of the radiative safe zone are set by two stellar fluxisopleths and the resulting shape is more complicated that that ofthe CHZ. Figure (2) illustrates a particular example: the gray regionis the radiative safe zone. Dashed lines indicate possible planetaryorbits that are not fit for life, while solid lines indicate safe orbits(we have approximated the planetary orbits as circles centered ineither star, or the barycenter). Notice that for an orbit to be safe,it must remain within the radiative safe zone at all times. In thisparticular case, there are safe circumprimary and circumsecondaryplanetary orbits, but no circumbinary ones.

In addition to the above complication, the separation betweenthe stars will vary for the general case of binaries with elliptical or-bits. First we will tackle the simpler case of circular orbit binaries.

2.1 Binaries in circular orbits

The star distances to the barycenter are given by::

rp = −(ms/M)r12, rs = +(mp/M)r12,

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Figure 2. Safe (continuous) and unsafe (dashed) planetary orbits. The ra-diative safe zone is the gray region. The planetary orbits are simply circlescentered on either star, or the center of mass of the system. In this case nocircumbinary safe orbits are possible.

where the p and s subindexes refer to the primary and secondarystars, m are the individual stellar masses and M is the total mass.Finally, r12 is the constant distance between the stars.

If we take the interstellar distance and total mass as units oflength and mass, we can write the previous relation as:

ηp = −q, ηs = +(1− q), (3)

where η is dimensionless distance and q = ms/M is the massfraction due to the secondary.

A similar scaling in luminosity can be accomplished adoptingLT as its unit. The dimensionless combined stellar flux is then:

I(ηx, ηy) =(1− λs)

(ηx + q)2 + η2y+

λs

(ηx + q − 1)2 + η2y, (4)

where (ηx, ηy) are the corresponding dimensionless Cartesian co-ordinates. Notice that I is completely set by the luminosity andmass ratios, its unit is LT /r

212.

2.1.1 The critical flux isopleth

The shape of the I–isopleths changes from a single (circumbinary)to a double (circumstellar) contour at the critical isopleth (see fig-ure 3). Its value is given by:

Ic(λ) =(1− λs)

1/3 + λ1/3s[

(1− λs)2/3 + λ2/3s − λ1/3

s (1− λs)1/3]2 (5)

Figure (4) shows the value of the critical flux isopleth as afunction of the secondary luminosity fraction. Notice that Ic doesnot depend on the mass ratio q. This is because the latter only shiftsthe isopleths on the orbital plane (see equations 3). The curve isalso symmetric with respect to λs = 1/2. This is because λ can

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�1.5 �1.0 �0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

x

y

q⇥0.5 ⇤⇥0.1

Figure 3. Various flux isopleths are shown for a binary with q = 0.5,λ = 0.1. The thick line is the critical isopleth. The plot origin is at thebarycenter frame and only the positive y axis is shown.

Figure 4. Critical isopleth flux value as a function of the secondary starluminosity fraction. Notice that Ic is symmetric with respect to λs = 0.5.

refer to either star (the more massive star is not necessarily the mostluminous one). So we drop the s subindex from now on.

2.1.2 The three different radiative zone configurations

Depending on the value of the dimensionless stellar fluxes that de-fine the CHZ (equation 1) with respect to the critical isopleth flux,the radiative safe zone may have three different configurations:

Configuration I: (Ic < Io). This corresponds to 2 separate cir-cumstellar radiative safe zones.Configuration II: (Io < Ic < Ii). This corresponds to a com-

bined case, where we find a circumbinary radiative safe zone withtwo inner holes around the stars.Configuration III: (Ii < Ic). This corresponds to a single radia-

tive safe zone with a single inner hole.

These configurations are illustrated in figure 5.In any of these configurations, safer planetary orbits are the

ones better contained within the safe radiative zone. In the moststrict sense, this means that we must consider their shape, whichintroduces an added complexity we will not get into. Since stable

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Configuration I Configuration II Configuration III

Figure 5. The three different radiative zone configurations. The illustrationis for a binary with λ = 0.4.

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-1.0 -0.5 0.0 0.5 1.00

0.2

0.4

0.6

hx

h y

Figure 6. Circumstellar boundaries of the rBHZ (configuration I). Theboundary isopleth is the dashed line. If this isopleth corresponds to the outerboundary, the largest safe orbit touches the isopleth on its ηx intersectionopposite the other star (left solid circle). If the isopleth is the inner boundary,the smallest safe orbit touches the isopleth at the opposite ηx intersection(right solid circle). The case shown is q = 0.3, λ = 0.2 and the isoplethdepicted is 1.0045Ic. The primary star, which in this example is also themost luminous, is the one on the left.

orbital planetary configurations in binaries have in general low ec-centricities (smaller to∼0.3, Pichardo et al. (2005)), we will definea radiative Binary Habitable Zone (rBHZ) as the largest circularannular region that can fit entirely within the radiative safe zone.We consider two cases: circumstellar and circumbinary zones. Or-bits that are safe for life will lay most likely within these zones.We should remember that these zones are defined by the radiativecondition only. In next section we will consider the dynamical con-dition and in section 4 we will combined both conditions to definethe combined binary habitable zone.

Now, a problem arises because the flux isopleths are not cir-cles (see figure 3), particularly close to the critical isopleth. Ourdefinition of BHZ implies that the largest safe orbit must be entirelyinscribed within the flux isopleth that defines the outer boundary ofthe safe zone. Similarly, the smallest safe orbit must be entirely cir-cumscribed outside the inner boundary flux isopleth. Note that thiscondition is not over the planetary orbit itself but just in order tofix the boundaries for habitability, in such way that it is possible toensure that a planet will remains all the time inside this zone.

The study can be separated depending on the configuration ofthe radiative safe zones, we now examine these conditions for eachcase.

2.1.3 Configuration I (Ic < Io)

In this case we have two separate circumstellar habitable zones.The orbital radius of the largest safe planetary orbit is given bythe smallest distance between the outer boundary isopleth and thecorresponding star. This ensures that the orbit will remain withinthe outer boundary isopleth.

Similarly, for the radius of the smallest, circumstellar safe or-bit, we must now consider the maximum distance from either starto the corresponding inner boundary flux isopleth, so the orbit re-mains outside it at all times.

The circumstellar isopleths are elongated along the ηx axisin the direction of the other star, while they are squashed in theopposite direction. This means that both, the outer and inner cir-cumstellar orbital radii (ros and ris, respectively) correspond tothe respective boundary isopleth intersections with the ηx axis (seefigure 6).

From equation (4), setting ηy = 0 and putting the star in ques-tion at the coordinate origin, we arrive at the following polynomial:

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1.0 10.05.02.0 3.01.5 7.0

0.001

0.002

0.005

0.010

0.020

0.050

0.100

!ê!c

dr isêris

Figure 7. Fractional error made using the CHZ criterion for a single star,instead of ris, as a function of the inner boundary flux value (in units of thecritical flux value). The solid line is for the most luminous star (λ = 0.9),while the dashed line is for the less luminous star (λ = 0.1).

Ibr4s − 2Ibr3s + (Ib − 1)r2s + 2λrs − λ = 0, (6)

where Ib is the dimensionless flux value that defines the rBHZboundary and rs is either, the outer circumstellar orbital radius ros,or the inner one ris, depending on the solution we choose. Unfor-tunately, although 4–degree polynomials can always be solved byradicals, the solution in this case is quite cumbersome. We havedecided instead, to list in table 1 some solutions. The first value ateach entry corresponds to ros (smallest value), the second to ris.We list results from the critical isopleth up to thrice its value only,because for larger values, the circumstellar boundaries are veryclose to the star and the single star CHZ criterion (equation (1))is sufficient. This is shown in figure 7, where the fractional errormade in using the CHZ criterion individually for each star, insteadof the true value obtained by solving equation (6), is shown as afunction of the flux value for the inner boundary. We can see thatfor values larger than those listed in table 1, the error drops wellbelow 1%.

2.1.4 Configuration III (Ii < Ic)

The considerations that define the largest and smallest boundariesare the same as for the previous case, except that this time we aredealing with boundary isopleths that surround both stars and therelevant extremal distances are with respect to the binary barycen-ter.

In this case there are two additional difficulties: the positionof the isopleths with respect to the barycenter depends on the massratio too and their form, close to the critical one, is nowhere nearcircular.

For the outer boundary, the relevant distance is that from thebarycenter to the closest point along the boundary isopleth. In gen-eral, this point is not along either axis (see figure 8).

From equation (4), we obtain the barycenter–isopleth distance.Minimizing this function, we get the orbital radius of the largestsafe orbit:

rob(q, λ, Io) =1

Io

√A1 +A2

2qf1(q), (7)

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Table 1. Some values of the radius of the largest/smallest safe circumstellar orbits rs (real solutions to equa-tion (6) for Ib > Ic)

Ib/Ic λ = 0.1 λ = 0.2 λ = 0.3 λ = 0.4 λ = 0.5 λ = 0.6 λ = 0.7 λ = 0.8 λ = 0.9

1 0.208/0.343 0.260/0.387 0.299/0.430 0.333/0.466 0.366/0.500 0.400/0.534 0.439/0.570 0.487/0.614 0.559/0.6751.001 0.208/0.316 0.260/0.378 0.299/0.421 0.333/0.457 0.366/0.491 0.400/0.525 0.439/0.561 0.487/0.605 0.559/0.6671.002 0.208/0.313 0.260/0.374 0.298/0.417 0.333/0.453 0.366/0.487 0.400/0.521 0.439/0.557 0.486/0.601 0.558/0.6631.005 0.208/0.306 0.259/0.367 0.298/0.410 0.332/0.446 0.365/0.480 0.399/0.513 0.438/0.550 0.486/0.593 0.558/0.6561.01 0.207/0.298 0.258/0.359 0.297/0.402 0.331/0.438 0.364/0.471 0.398/0.505 0.437/0.541 0.484/0.585 0.556/0.6481.02 0.206/0.288 0.257/0.348 0.296/0.390 0.329/0.426 0.362/0.459 0.396/0.493 0.435/0.529 0.482/0.573 0.553/0.6361.05 0.202/0.269 0.253/0.327 0.291/0.368 0.324/0.403 0.357/0.436 0.390/0.469 0.428/0.506 0.475/0.549 0.545/0.6121.1 0.197/0.248 0.246/0.305 0.284/0.345 0.316/0.379 0.348/0.411 0.381/0.444 0.418/0.479 0.464/0.522 0.533/0.5841.2 0.187/0.223 0.234/0.277 0.271/0.314 0.302/0.347 0.333/0.377 0.364/0.409 0.400/0.443 0.444/0.485 0.510/0.5461.5 0.164/0.181 0.207/0.228 0.240/0.262 0.269/0.291 0.296/0.319 0.325/0.347 0.357/0.378 0.396/0.417 0.456/0.4732 0.139/0.147 0.177/0.187 0.206/0.216 0.231/0.242 0.255/0.266 0.280/0.291 0.308/0.318 0.343/0.352 0.394/0.4023 0.112/0.115 0.143/0.147 0.167/0.171 0.187/0.191 0.207/0.211 0.228/0.232 0.251/0.255 0.279/0.283 0.321/0.324

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-1.0 -0.5 0.0 0.5 1.00

0.2

0.4

0.6

hx

h y

Figure 8. Circumbinary boundaries of the rBHZ (configuration III). Theboundary isopleth is the dashed line and the barycenter is at the coordinateorigin. If this isopleth corresponds to the outer boundary, that largest safeorbit touches the isopleth at the point where the barycenter to isopleth dis-tance is minimum (inner circle). If the isopleth is the inner boundary, thesmallest safe orbit touches the isopleth at the ηx intersection farthest awayfrom the barycenter (outer circle). The binary parameters are the same asthose in figure (5), except that the isopleth depicted is 0.779Ic

where rob denotes the outer circumbinary orbital radius and Io isthe flux value that defines the outer boundary isopleth (in this caseIo < Ic). The auxiliary quantities are given by:

A1 = 2[Ioqf1(q)]2 + Iof22 (q)

√λqf1(λ)f1(q), (8)

A2 = Ioqf1(q)[√λf1(λ)

qf1(q)− 2[qf2(λ) + f1(λ)]

], (9)

and

f1(x) = x− 1, f2(x) = 1− 2x (10)

Equation (7) is symmetric around λ = 1/2. This is expected,since a reflexion around the ηy axis of figure (8) leaves the shortestbarycenter to isopleth distance unchanged.

Notice that the radius of the largest safe circumstellar orbitdoes not depend on q, whereas that of the circumbinary orbit does.As we mentioned, this is because the position of the isopleths isfixed with respect to the stars, but not with respect to the barycenter(equation 3).

For the inner boundary, the proper orbital radius is the distancefrom the barycenter to the farthest away ηx axis intersection of theboundary isopleth (figure 8). The easiest way to find this distance isto solve equation (4) for ηy = 0, which leads to the same polyno-mial as in case I (equation 6). But now (Ib < Ic), the polynomialhas only two real roots, which are the distances from either star tothe boundary isopleth (r?). To find the final inner circumbinary or-

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10-4 0.001 0.01 0.1 1

0.01

0.02

0.05

0.10

0.20

!ê!c

dr ibêrib

Figure 9. Fractional error made using the CHZ criterion for a single star atthe barycenter with the total binary luminosity, instead of rib, as a functionof the inner boundary flux value (in units of the critical flux value). Theλ = 0.9 case is shown.

bital radius, we now add the respective star to barycenter distance(equation 3) to each root and compare the resulting barycenter toisopleth distances. The largest one is rib.

As before, the solution is rather complicated and some valuesof r? are shown in table 2. For distant boundaries (small valuesof Ii/Ic), the isopleths become circular and converge to the CHZcriterion applied using the total binary luminosity. This can be seenin figure (9), where the fractional error made using the CHZ insteadof the rBHZ criterion is plotted for the λ = 0.9 case. Again, forvalues beyond those listed in table 2 the error is very small.

In this case it may be that rib > rob, which means that nosafe circular orbits centered in the barycenter exist. However, anon–circular and sufficiently elongated orbit may remain within therBHZ, but this is beyond this study.

2.1.5 Configuration II (Io < Ic < Ii)

This is the most complicated configuration to compute, since wenow have the possibility of both, circumstellar and circumbinarysafe zones. This is a mixture of the previous two cases and theyhave to be dealt separately.

For the circumstellar zone, we use the procedure used forcase I, with Ib = Ii, to interpolate values of ris for each star fromtable 1. For the outer edge, we use the procedure of case III withIb = Io, to obtain from equation (7) a value that we will interpret

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Table 2. Some values of the star to outer boundary isopleth r? (negative real solution to equation (6) for Ib < Ic)

Io/Ic λ = 0.1 λ = 0.2 λ = 0.3 λ = 0.4 λ = 0.5 λ = 0.6 λ = 0.7 λ = 0.8 λ = 0.9

0.99 0.210 0.261 0.300 0.335 0.368 0.403 0.441 0.489 0.5620.90 0.222 0.276 0.316 0.352 0.387 0.423 0.463 0.514 0.5890.80 0.239 0.295 0.338 0.375 0.412 0.450 0.492 0.546 0.6260.60 0.287 0.349 0.397 0.439 0.480 0.523 0.571 0.632 0.7230.40 0.379 0.446 0.500 0.549 0.597 0.648 0.705 0.778 0.8880.20 0.635 0.695 0.754 0.812 0.872 0.938 1.014 1.111 1.2620.10 1.084 1.101 1.151 1.213 1.282 1.363 1.461 1.590 1.7950.05 1.796 1.736 1.759 1.814 1.889 1.984 2.106 2.275 2.5520.01 4.978 4.624 4.512 4.508 4.574 4.701 4.900 5.211 5.7690.005 7.392 6.833 6.623 6.571 6.624 6.767 7.017 7.427 8.1880.001 17.61 16.20 15.59 15.34 15.34 15.54 15.99 16.80 18.400.0005 25.27 23.24 22.33 21.93 21.88 22.13 22.72 23.84 26.070.0001 57.60 52.93 50.76 49.75 49.51 49.95 51.16 53.53 58.40

Out[626]=

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.0

-0.5

0.0

0.5

1.0

Figure 10. Circumbinary safe orbits in configuration II. The dashed lineis the critical isopleth and the shaded area is the radiative safe zone. Theextremal safe circumbinary orbits, which define the boundaries of the cir-cumbinary rBHZ are shown (continuous line circles). The example shownis for q = 0.44, λ = 0.375 and Io/Ic = 0.179, Ii/Ic = 1.592.

as ros. Notice that for the outer circumstellar edge we are using anisopleth that is not circumstellar. This is because a safe circumstel-lar orbit may reach beyond the critical isopleth1 (see figure 12). It isobvious that if the resulting ros reaches all the way to the inner edgeof the circumstellar safe zone of the other star, then the smaller ofthe two distances should be taken, since a safe circumstellar orbitshould not get into the unsafe region of the other star.

For the circumbinary region, we apply the procedure ofcase III with Ib = Io for the outer edge. For the inner edge wehave to deal again with the possibility of safe orbits that cross thecritical isopleth (see figure (10)). The inner circumbinary orbitalradius is given by the largest of the two barycenter to inner bound-ary isopleth ηx intersections farther away in the direction to eachstar. These distances are given by the individual ris for each star(which we computed already), plus their respective star distance tothe barycenter.

1 We should remember that the Roche lobes, which are the regions of dy-namical influence of each star, are not the same as the circumstellar regionsdefined by the critical flux isopleth.

2.1.6 Procedure to obtain the habitability region

Tables 1 and 2, together with equation (7) and auxiliary relations,define the orbital radii of the largest and smallest circumstellar orcircumbinary safe edges.

The detailed procedure to find the limits of the rBHZ is thenas follows:

(i) From the binary components individual masses and lumi-nosities, obtain the dimensionless mass fraction q and luminosityfraction λ, the latter for both stars.

(ii) Get specific boundary radiative stellar flux values Io, Ii (e.g.Kopparapu et al. (2013)) and convert them to the dimensionlesssystem we use here: Ii, Io (normalized to total binary luminosityper square semimajor axis).

(iii) Given λ (either star), use equation (5) to find the value ofthe critical flux for this case: Ic.

(iv) If Io > Ic, proceed to configuration I; if Ii < Ic, proceedto configuration III; else, go to configuration II.

(v) Configuration I: Two separate radiative circumstellar hab-itable zones. Use table 1 to interpolate the ros and ris edges radiifor each star using the appropriate value of Ib.

(vi) Configuration II: One radiative circumbinary habitablezone with two circumstellar inner edges. We have to solve the cir-cumstellar and circumbinary zones separately.

(a) Circumstellar zones: Use table 1 to interpolate the risradii for each star using Ib = Ii.

(b) Circumbinary zone: Use equation (7) to get the outermostradius rob. For the smallest radii, add to the ris previously com-puted for each star their respective star to barycenter distances,the largest of the two is the radius of the innermost safe circumbi-nary edge.

(vii) Configuration III: One radiative circumbinary habitablezone whose inner edge surrounds both stars. For the outer bound-ary, use equation (7) to compute rob. For the inner boundary, usetable 2 to compute the distances of each star to its closest ηx inter-section of the inner boundary isopleth r?. To each add the barycen-ter to respective star distance using equation (3). The largest of thetwo is rib.

Figure (11) shows the result for a specific example. We havea primary G2V star with 1M� and 1L�, while the secondary isa G8V star with 0.8M� and 0.6L�. The stars are assumed to be4AU apart. We have used the stellar flux limits of Kopparapu etal. (2013) for a G2 star: Io = 0.53 and Ii = 1.10L�/AU

2,which correspond to the CO2 condensation and water loss limits,

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Figure 11. The critical flux isoplet (thick, dashed line) and radiative binaryhabitability zones (shaded areas) for the first worked example. The limitingsafe orbits are shown (thick, continuous lines). The frame is centered in thebarycenter and the unit of length is the binary separation, which in this caseis 4 AU .

Out[960]=

-1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

Figure 12. Same as figure (11), but in this case the binary separation hasshrunk to 3 AU (taken as unit length in the plot). Notice that the largestcircumstellar orbits reach outside the critical isopleth.

respectively. In this case it is clear that safe orbits can exist withintwo circumstellar zones.

Figure (12) shows the result for the same binary, but now witha separation of 3AU . By putting the stars closer together, the cir-cumstellar safe zones have merged into a single, case II habitablezone. In this case, the only safe orbits lie between the ris and theouter boundary isopleth. No circumbinary safe orbits are possible.

2.2 Binaries in eccentric orbits

Once the case of the binary in a circular orbit has been solved, theeccentric orbit case is rather simple. We use the circular orbit proce-dure twice: at periastron and apoastron, and compare the resultinglimits. For the smallest safe orbit, we take the largest of the com-puted values for both orbital extrema. For the largest safe orbit, wetake the smallest of the respective values. This guarantees that afixed, circular, planetary orbit will remain within the safe zone forall orbital phases. Again, if the resulting outer limit is smaller thanthe inner limit, no safe orbit is possible.

We must stress again that, although this procedure does nottake into account explicitly non–circular orbits for planets, nor doesit take into account the possible variation in their shape as a func-tion of binary orbital phase (if any), we claim that planetary orbits

in binary eccentric systems will not develop readily, highly eccen-tric stable orbits (i.e. e . 0.3). Furthermore, these deforming ef-fects on orbits in binary systems, may arise close to the criticalisopleth, however, it does not necessarily coincides with the loopgap between circumstellar and circumbinary stable orbits.

3 STABILITY CONDITION FOR HABITABLE ZONES INBINARY STARS

In this paper we have employed the criteria of Pichardo et al.(2005) and Pichardo et al. (2009) for circumestellar and circumbi-nary stable orbits respectively. The same approach was used inJaime et al. (2009), where stability zones were studied for binariesof the solar neighborhood. We show in this paper why this stabilitymethod results better in searching habitable regions for planets.

Unlike the fundamental work of Holman & Wiegert (1999),that represents an excellent empirical approximation to the stabil-ity problem, invariant loops provide an exact solution for stableorbits in ideal (isolated) binary systems at any binary eccentricity.In this manner, rather than orbits that keep stable for some smallfraction of the binary star life, to ensure life development, muchmore time than a small fraction of the binary life is needed. Withinvariant loops, numerical integrations are used to identify them inphase space, their existence and moreover, their stability, are sup-ported by integrals of motion in the extended phase space. As longas the orbital parameters of the binary are not changed, the sta-bility of the loops is guaranteed without having further numericalintegrations (see Pichardo et al. 2005, Section 2). Consequently,due to the methodology, Holman & Wiegert´s approximation over-estimates the available stable regions since their stability criteriadepends on a given time of integration that the particles keep inorbit (about 104 periods), which results in a very relaxed criteriawhen looking for strict stability, needed for life emergency and itsdevelopment. Furthermore, invariant loops show in the circumbi-nary discs cases, that there is a shift of the stable region that can beconsiderable for high eccentricities, compromising the intersectionbetween stable regions for planets and habitable regions (as is thecase, for example of Kepler 16). The shift can not be detected withthe Holman & Wiegert method, this was detected theoretically andpresented in Pichardo et al. 2008 paper. We consider and computedthis shift in the present work.

In our approach however, we do not consider multiple stars,nor stars that have evolved away from the main sequence, or plan-ets in highly eccentric orbits. In the case of multiple stars, this isfirst because it is not straightforward to extend the loop formalismto that case, and furthermore, multiple systems result generally inunstable systems, unless they are hierarchical (v.g. triple systemswhere a very close binary star with a far companion as the AlphaCentauri system and Proxima Centauri seem to be, or a system witha double binary where both binary systems act like a whole binarysystem since they tend to be extremely separated), and in that case,these systems can be reduced to regular separated binary systems ata good approximation. On the same direction, in the general case ofmultiple stars, stable regions for planetary orbits would be severelycompromised, and even if stable regions would exist, finding themwould be a challenging task. Thus, if we are not able to calculatestable regions, it is not of importance to find the correct irradi-ance zones because we still would not be able to establish habitablezones (i.e. the combination of stability and stellar irradiance).

Regarding the type of star, although our method allows us tocalculate radiative safe zones for any type of star, in our specific

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study for solar neighborhood binaries, we only calculated habitablezones for main sequence stars since those out of this “life stage”,would likely be inhospitable for life.

Finally, we do not include studies of planets on very eccentricorbits (i.e. ep & 0.3), in our experience and work with stabilityzones in binaries constructed out of invariant loops, we have foundthat their probability of survival is highest, for example, either onplanets in a circumbinary disc for the case of very close binary sys-tems (where the circumbinary material feels the system almost as asingle star), or in circumstellar discs, for very open binaries (whereboth stars act almost as single stars). However, this is not the gen-eral case, in any other binary system in between, stable regions forplanetary orbits with high eccentricities (ep & 0.3), the reductionof the available phase space for planets to settle down, and conse-quently, of their possibilities to be stable for long timescales can beconsiderable (this is an ongoing study that will be presented in afuture paper).

Under this approach stable zones are defined by invariantloops (see Pichardo et al. (2005) and Pichardo et al. (2009) fordetails). The radius for the most exterior dynamically stable orbitaround each star is given by:

Ri = REggi

(0.733(1− e)1.2q0.07

), (11)

In a similar manner, the inner viable radius for circumbinary stableorbits is:

RCB ≈ 1.93a(1 + 1.01e0.32

)(q(1− q))0.043 , (12)

where subindex i represents each star primary (i = 1) and sec-ondary (i = 2), a is the semimajor axis of the binary, e the eccen-tricity and q = q2/(q1 + q2) is the mass ratio (see eq. (3)).

Finally, in equation (11),REggi is the approximation of Eggle-

ton Eggleton (1983) to the maximum radius of a circle within theRoche lobe, given by,

REggi =

0.49aq2/3i

0.6q2/3i + ln(1 + q

1/3i )

, (13)

in this equation qi is defined by q1 = M∗1/M∗2 and q2 =M∗2/M∗1.

For the general eccentric case, one must consider a shift of thecenter of the minimum radius for stable circumbinary orbits, RCB

( eq. (12)). This shift is given by,

Rsh = −3.7 a e0.8 (0.5− q) [q(1− q)]1/4 . (14)

It is important to stress that the orbits defined by equa-tions (11) and (12), represent stable orbits formed by non self-intersecting loops, where gas and planets could settle down in longterm basis.

4 THE COMBINED HABITABILITY CONDITION:SOME PARTICULAR CASES.

As we have mentioned, the combination of the radiative and stabil-ity conditions is one of the goals of the present paper. In order toshow how this approach allows us to find candidates in the searchfor habitable planets, in this section we apply our method to 64binary systems of the solar neighborhood, with known orbital pa-rameters, and main sequence stars. Most of the cases are eccentricbinaries, thus we apply our procedure (both, radiative and stability

conditions) at periastron and apoastron. Once we have these twocalculations, we compare both and take, conservatively, the mostrestricted circular edge. It is worth to mention that it is importantto compute it in both locations because we do not know a prioriin which case the binary will be located (i.e., if the habitable zonewill be circumstellar or circumbinary). Given this, the restriction atperiastron or apoastron acts in different ways for configurations I,II or III, and the case can even change from one to another, and thisshould be taken into account.

Most of binaries considered in this paper are taken from Jaimeet al. (2009), which correspond to binaries with stars in the mainsequence. We have applied the classical stellar luminosity-massfunction Cox (2000) in order to obtain the luminosity for each star.Orbital parameters are the same as the ones considered in Jaime etal. (2009).

Another important issue that must be considered, is the cir-cumbinary disc center shift, produced by eccentric binary systems(eq.(14)). In high eccentric cases this shift might become relevantwhen calculating the circumbinary orbits inside the habitable zone.

Table 3 shows the results for the 64 objects at periastron, col-umn 1 is the Hipparcos name, column 2 is an alternative name,column 3 shows the particular habitable case resulting for the bi-nary, columns 4 and 5 contain the inner and outer habitable radiifor the primary star and columns 6 and 7 are the same but aroundthe secondary star. Finally columns 8 and 9 provide the inner andouter habitable radii in the circumbinary case. Table 4 shows thesame as Table 3 but at apoastron.

Table 5 shows orbital parameters for each binary, columns 1and 2 are the HIP name and alternative name, column 3, 4, 5 and6 are semimajor axis, eccentricity, primary and secondary massesrespectively. Column 7 shows the outer stability radius around theprimary star while column 8 shows the same but for the secondarystar of the binary. Column 9 shows the inner circumbinary stabilityradius for the object and column 10 gives the shift for the circumbi-nary stability radius. Finally column 11 provides the reference fororbital parameters used in this paper.

Table 6 shows the cases where all criteria are satisfied together,i.e. objects where the intersection of habitability at periastron andapoastron is not empty. For the objects in this table we can observesome habitability zone well defined at periastron and apoastron atthe same time and also inside of the stability zone given by equa-tions of section 3. In circumbinary cases the shift was taken intoaccount.

In this section three particular cases are considered in order toshow how the approach is implemented.

HIP 1995HIP 1995 has a semimajor axis of a = 0.54AU with an ec-centricity e = 0.33, stellar masses are M∗1 = 1.13M� andM∗2 = 0.45M�. We have included habitable zones at perias-tron and apoastron (tables 3 and 4), this two separate calculationswill restrict the effective habitable zone, which is located just ina circumbinary position. At Periastron we have R∗1

P (inner) =1.49AU , R∗1

P (outer) = 2.22, and at apoastron R∗1A (inner) =

1.59AU , R∗1A (outer) = 2.2. The dynamical stability is given

by the minimum radius of the circumbinary stable zone, follow-ing equation (12) this starts at RCB = 1.66AU . As an additionalrestriction, the shift should be considered, its value in this case isimportant because it change the viable zone defined under our ap-proach, Rshift = −0.11AU . Figure 13 shows all of these esti-mations, black (semi) dots are the stars located at periastron while(semi) dots in gray show them at apoastron. Gray disks are the hab-itable zones, calculated for both cases. Dark gray provides the in-

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- 2 -1 0 1 20.0

0.5

1.0

1.5

2.0

x H AUL

yHA

UL

Figure 13. Effective habitable zone (radiative and stability conditions) forHIP 1955. Black dots are the stars at periastron, gray dots at apoastron. Theshaded region is the radiative habitable zone, with darker gray indicatingthe intersection between these two zones at periastron and apoastron. Theblue dashed circle represents the minimum dynamical stable orbit aroundthe binary and the red one is the same but considering the effect because theshift.

-1 - 0.5 0 0.50

0.2

0.4

0.6

x H AUL

yHA

UL

Figure 14. Effective habitable zone for HIP 80346. Black dots are the starslocated at periastron, gray disks shows the habitable zone, darker gray diskillustrate the intersection of habitable at periastron and apoastron. Red cir-cles shows the maximum dynamical stable orbit around each star. We canobserve only secondary star have a complete effective habitable zone.

tersection of the BHZ, the effective habitable zone for the binary.Dashed blue circle is the minimum dynamically stable circumbi-nary orbit, and the red one is the same but with the shift consid-ered, this is the inner border of the viable stable orbit. Taken in toaccount all of these criteria is how we decide where is located thecompletely viable BHZ.

HIP 80346This is a particular case, in this object we can found an effectivehabitability within the dynamical stability zone just around the sec-ondary star. Semimajor axes of this binary is a = 2.07AU , ec-centricity e = 0.67, nevertheless stellar masses are very small,M∗1 = 0.5M� andM∗2 = 0.13. The main restriction to habitabil-ity is because the eccentricity of the binary. The dynamical stabilityregion is given by equation 11, by using this relation we obtain themaximum stable orbit radius around each star, R∗1

ce = 0.18AUand R∗2

ce = 0.1AU . Figure 14 shows both stars at periastron, thegray disks are the habitable zones, where the darker gray allows tosee the habitable zone with the value at apoastron considered. Redcircle around each star provides the maximum radius for dynami-cal stable orbits. We can observe that for the primary star habitablezone is out of dynamical stability, but in the secondary we can ob-serve habitability within stable zone. Figure 15 shows a zoom forthe secondary star of this object.

0.2 0.3 0.4 0.50

0.05

0.1

0.15

x H AUL

yHA

UL

Figure 15. Zoom of secondary star of HIP 80346. This star can provide acomplete effective habitable zone. Darker gray disk is where we can expectan habitable planet.

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

x

y

Figure 16. A pathological case, HIP 76852: Dashed lines show the bound-aries of the radiative safe zone. This is a type III configuration. However, asshown by the red circles, no circumbinary BHZ is possible despite having anon-zero radiative safe zone.

HIP 76852

HIP 76852 is a peculiar configuration III case, because we cannot find a well defined BHZ (the inner boundary has a radius largerthan the outer one), and yet, it has a non-zero radiative safe zone.Figure 16 shows this case (in units of the separation at periastron),black dots are primary (left) and secondary (right) stars located atperiastron, interior dashed line is the flux isoplete that define theinner habitable zone, the exterior dashed line is the same but forthe outer boundary. The inner red circle is what we have definedas the minimum circunscrite circle to the outer habitable boundaryand the exterior red circle is the circle that circunscribe the innerflux isoplete boundary for habitability. This way we can observe itis not possible to have both conditions satisfied at the same time,nevertheless an habitable zone is well defined around the binary.This is a very important restriction because, in case we have thiskind of objects we can not tell for sure if a planet can be hostedwithin this zone, the eccentricity that the planet must have in orderto be all the time inside the habitable zone could be very high oreven the shape of the orbit could be non physical. So this case showus it is not always possible to find a well behaved habitable zone,where some planet can settle.

5 DISCUSSION AND CONCLUSIONS

In this work we have constructed a straightforward and clear for-mulation to calculate regions for habitable planets in binary stellarsystems. To this purpose, we search for two general restrictionsassuming in principle Earth-like planets: a) the planet must be lo-cated in a region of orbital stability (and approximately circular

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orbit), and b) the planet is located at a position, such as it reachesthe correct host star energy, to permit the existence of liquid wateron its surface Kopparapu et al. (2013). Some other particular re-strictions, as the ones proper of the intrinsic characteristics of theplanet for example, can readily be addressed to this formulation asmultiplicative factors.

Regarding the fundamental test for orbital stability to developlife, in this paper we employ the criteria of Pichardo et al. (2005)and Pichardo et al. (2009) for stable orbits in binary systems. Thisstability method results better in searching habitable regions forplanets. Indeed, although the empirical approximation of Holman& Wiegert was an excellent and fundamental first approximationto the stability problem, their method is rather based on a detailedtrial and error approximation, that overestimates by construction,the available stable regions since their stability criteria depends ona given time of integration that particles keep in orbit (about 104

periods), which results on a relaxed criteria when looking for strictstability, needed for life emergency and developing. Instead, wehave employed the invariant loops method that searches for the ex-act solution for stable orbits in binary systems at all binary eccen-tricities.

On the other hand, invariant loops show in the circumbinarydiscs cases, that there is a shift of the geometric center of the sta-ble region (disk) that can be considerable for high eccentricities,compromising the intersection between stable regions for planetsand habitable regions in the simple sense of the irradiance (as is thecase, for example of Kepler 16). The shift can not be detected withother method than the exact solution provided by the invariant loopstool, this was detected theoretically and presented in Pichardo etal. 2008 paper, we are considering this for planets in circumbinarydisks in our calculations for binaries of the solar neighborhood.

It is worth mentioning that in our approach to calculate hab-itable zones in binary stars, for different life likelihood consider-ations, such as orbital stability and irradiation, we are not consid-ering multiple stars, or stars out of their main sequence or planetson highly eccentric orbits. In the case of multiple stars, althoughan extension of this work to any multiple star systems would bestraightforward for the irradiance calculations, in the majority ofcases these systems result in unstable ones, unless they are hier-archical (v.g. triple systems where a very close binary star with afar companion as Alpha Centauri and Proxima, or a double binarywhere both binary systems act like single stars in one larger binarysystem). On the other hand, there is no straightforward method tofind stable orbital zones in multiple stellar systems, needed to cal-culate formally habitable zones.

Although our method is aplicable to all kind of stars, we onlycalculated habitable zones for the solar neighborhood main se-quence stars, since stars out of this life stage would likely be in-hospitable for life. In the case of planets on very eccentric orbits(e&0.3), the probability of survival is diminished due to the pres-ence of the companion that reduces severely the available phasespace for planets to settle down (this is a detailed study that will bepresented in a future work). On the other hand, what induces eccen-tricity on a planet in a binary star?, external factors are stellar en-counters, but to affect a very small disc (truncated by the binary), avery close stellar encounter (with a third star) must be taking place(∼3 times the disk radius at least, Jimenez-Torres et al. (2011)),likely affecting the binary stability itself, this results in a non stablesituation for habitability. Or an internal factor, for example a reso-nance that is able to induce secularly eccentricity in the planetaryorbits, however that is also an unstable situation for planets, sinceresonances tend to increase rapidly eccentricities on orbiting bod-

ies, wiping out entire disk regions. Other possibility, a giant planeton eccentric orbit that induces eccentricity in terrestrial planets, aclearly difficult situation in terms of stability.

Regarding irradiance zones. We have defined three zoneswhere the binary star provides the necessary energy for habitability:(I) the zone around each star, (III) the zone around the hole binaryor (II) in a mixture of this two zones. In this work, we consider a“habitable environment”, the intersection of one of these zones andthe allowed dynamical zone for stable orbits.

Taking into account both restrictions, from a binary sampleof main sequence stars, with known orbital parameters of the solarneighborhood (64), we have selected 36 candidates (56 % of ouroriginal sample), as plausible candidates to host habitable planets.We present this table together with three particular and interestingexamples in detail: HIP 1995, HIP 80346 and HIP 76852.

We find, from our sample of candidates, that none allows plan-ets inside the BHZ defined by the configuration II (circumbinarydiscs). This is because in all cases, the system allows habitabilitytoo close to the binary, where the stability restriction becomes veryimportant, making impossible for all cases in our sample, to host aplanet there. Although a greater sample is necessary to produce afinal conclusion on the solar neighborhood, this small first sampleis useful to statistically elucidate the possibilities of finding habit-ability on binaries of the solar neighborhood, and the possibilitiesfor circumbinary discs seem reduced.

Programming our formulation for habitable zones (from theirradiance point o view) is rather simple, however software to com-pute numerically the size of habitable zones for binary systems isavailable from the authors.

We thank the anonymous referee for a very throrough re-view of our manuscript and suggestions that resulted in a clearerand deeper exposition. We acknowledge financial support fromUNAM/DGAPA through grant IN114114.

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HIP Alter. name CaseP R∗1P inner R∗1

P outer R∗2P inner R∗2

P outer R∗∗P inner R∗∗

P outer[AU] [AU] [AU] [AU] [AU] [AU]

– δ Equ 3 - - - - 4.51 6.49– V821 Cas 3 - - - - 5.04 8.55

1349 HD 1273 3 - - - - 1.26 1.781955 HD 2070 3 - - - - 1.49 2.222941 ADS520 1 0.55 0.95 0.55 0.95 - -5842 HD 7693 1 0.87 1.50 0.78 1.34 - -7078 HD 9021 3 - - - - 1.78 2.707751 HD 10360 1 0.66 1.14 0.63 1.08 - -7918 HD 10307 1 0.71 1.22 0.03 0.05 - -8903 HD 11636 3 - - - - 4.68 7.6811231 HD 15064 3 - - - - 1.35 2.0212062 HD 15862 3 - - - - - -12153 HD 16234 3 - - - - 128.82 219.8512623 HD 16739 3 - - - - 2.53 4.1614954 HD 19994 1 1.92 3.30 0.15 0.25 - -18512 HD 24916 1 0.15 0.25 0.04 0.06 - -20087 HD 27176 2 3.50 4.44 1.41 2.28 - -20935 HD 28394 3 - - - - 2.02 3.2524419 HD 34101 2 0.89 1.53 0.07 0.18 1.37 1.27∗

30920 1 0.06 0.10 0.01 0.02 - -33451 HD 51825 3 - - - - - -34164 HD 53424 3 - - - - 1.77 2.1739893 3 - - - - 1.50 1.5344248 1 2.19 3.75 0.90 1.60 - -45343 1 0.31 0.54 0.30 0.52 - -56809 HD 101177 1 3.86 6.63 1.95 3.34 - -63406 HD 112914 2 0.76 0.96 0.11 0.30 0.98 1.1064241 2 1.90 3.17 1.47 2.48 - -64797 1 0.60 1.03 0.31 0.54 - -67275 HD 120136 1 1.92 3.30 0.19 0.33 0 -67422 1 0.58 1.00 0.48 0.82 - -72848 HD 131511 3 - - - - 1.05 1.5673440 HD 133621 3 - - - - 1.27 1.8375379 HD 137502 3 - - - - 1.84 2.8676852 HD 140159 3 - - - - - -79101 HD 145389 3 - - - - 11.98 18.5180346 1 0.29 0.50 0.03 0.05 - -80686 HD 147584 3 - - - - 1.23 1.9082817 HD 152771 1 0.36 0.62 0.14 0.25 - -82860 HD 153597 3 - - - - 1.59 2.4284425 2 1.73 2.82 0.97 1.61 - -84720 HD 156274 1 0.69 1.19 0.26 0.44 - -84949 HD 157482 3 - - - - 7.35 11.0186400 HD 1360346 3 - - - - 0.70 0.9886722 HD 161198 3 - - - - 1.03 1.5487895 HD 163840 3 - - - - 1.61 1.8891768 HD 173739 1 0.18 0.31 0.14 0.24 - -93017 ADS 11871 1 2.86 4.93 2.64 4.56 - -93825 1 1.72 2.95 1.66 2.87 - -95028 HD 181602 3 - - - - 2.21 3.2895575 HD 183255 3 - - - - 0.86 1.1098001 HD 188753 2 1.88 3.16 1.43 2.43 - -99965 HD 193216 3 - - - - 1.31 1.42

109176 HD 210027 3 - - - - 1.83 2.99111170 HD 213429 3 - - - - 1.72 2.21113718 HD 217580 3 - - - - 0.75 1.00116310 HD 221673 1 4.06 6.99 4.06 6.99 - -116727 HD 222404 1 2.62 4.50 0.20 0.35 - -

Kepler 16 3 - - - - 0.58 0.85Kepler 34 3 - - - - 1.65 2.81Kepler 35 3 - - - - 1.76 2.75Kepler 38 3 - - - - 0.64 0.99Kepler 47 3 - - - - 0.12 0.21Kepler 64 3 - - - - 2.60 4.44

Table 3. Radiative habitable zones at Periastron. Columns are: (1) HIP name, (2) Alternative name (some cases), (3) rHZ case described in section 2.1.2, (4)Inner rHZ for primary star, (5) Outer rHZ for primary star, (6) Inner rHZ for secondary star, (7) Outer rHZ for secondary star, (8) Inner rHZ for binary zoneand (9) Outer rHZ for binary zone. Note: rHZ - Radiative Habitable Zone.

c©—- RAS, MNRAS 000, 1–??


HIP Alter. name CaseA R∗1A inner R∗1

A outer R∗2A inner R∗2

A outer R∗∗A inner R∗∗

A outer[AU] [AU] [AU] [AU] [AU] [AU]

– δ Equ 3 - - - - - -– V821 Cas 3 - - - - 5.04 8.55

1349 HD 1273 3 1.16 1.82 0.50 0.79 - -1955 HD 2070 3 - - - - 1.59 2.202941 ADS520 1 0.55 0.95 0.55 0.95 - -5842 HD 7693 1 0.87 1.50 0.78 1.34 - -7078 HD 9021 3 - - - - 1.91 2.677751 HD 10360 1 0.66 1.13 0.63 1.08 - -7918 HD 10307 1 0.71 1.22 0.02 0.04 - -8903 HD 11636 3 - - - - 5.01 7.66

11231 HD 15064 3 - - - - 1.48 2.0012062 HD 15862 3 1.00 1.70 0.25 0.47 - -12153 HD 16234 3 - - - - 129.76 219.8212623 HD 16739 3 - - - - 3.09 4.0314954 HD 19994 1 1.92 3.30 0.15 0.26 - -18512 HD 24916 1 0.15 0.25 0.04 0.06 - -20087 HD 27176 2 3.24 5.50 1.10 2.00 - -20935 HD 28394 3 - - - - 2.17 3.2124419 HD 34101 2 0.89 1.53 0.06 0.15 - -30920 1 0.06 0.10 0.01 0.02 - -33451 HD 51825 3 2.72 4.67 1.73 3.02 - -34164 HD 53424 3 - - - - - -39893 3 1.02 1.71 0.37 0.67 - -44248 1 2.18 3.74 0.89 1.55 - -45343 1 0.31 0.54 0.30 0.52 - -56809 HD 101177 1 3.86 6.63 1.95 3.35 - -63406 HD 112914 2 0.75 1.28 0.07 0.14 - -64241 1 1.79 3.09 1.35 2.34 - -64797 1 0.60 1.03 0.31 0.54 - -67275 HD 120136 1 1.92 3.30 0.19 0.33 - -67422 1 0.58 1.00 0.48 0.82 - -72848 HD 131511 3 - - - - 1.21 1.5273440 HD 133621 3 1.15 1.97 0.06 0.46 1.38 1.7975379 HD 137502 3 - - - - 2.24 2.7776852 HD 140159 3 4.43 7.34 4.35 7.21 - -79101 HD 145389 3 - - - - 12.53 18.4680346 1 0.29 0.50 0.02 0.04 - -80686 HD 147584 3 - - - - 1.23 1.9082817 HD 152771 1 0.36 0.62 0.14 0.24 - -82860 HD 153597 3 - - - - 1.63 2.4284425 1 1.61 2.78 0.82 1.43 - -84720 HD 156274 1 0.69 1.19 0.26 0.44 - -84949 HD 157482 3 - - - - 9.28 10.3886400 HD 1360346 3 - - - - 0.76 0.9686722 HD 161198 3 0.97 1.66 0.14 0.25 - -87895 HD 163840 3 1.11 1.86 0.58 1.01 - -91768 HD 173739 1 0.18 0.31 0.14 0.24 - -93017 ADS 11871 1 2.82 4.87 2.60 4.49 - -93825 1 1.71 2.94 1.66 2.85 - -95028 HD 181602 3 - - - - 2.37 3.2595575 HD 183255 3 - - - - 0.92 1.0798001 HD 188753 2 1.79 3.09 1.33 2.30 - -99965 HD 193216 3 - - - - 1.38 1.38109176 HD 210027 3 - - - - 1.83 2.99111170 HD 213429 3 - - - - - -113718 HD 217580 3 0.64 1.11 0.04 0.09 - -116310 HD 221673 1 4.05 6.97 4.05 6.97 - -116727 HD 222404 1 2.62 4.50 0.19 0.32 - -

Kepler 16 3 - - - - 0.59 0.85Kepler 34 3 - - - - 1.67 8.90Kepler 35 3 - - - - 2.44 3.80Kepler 38 3 - - - - 0.71 1.09Kepler 47 3 - - - - 0.11 0.19Kepler 64 3 - - - - 1.18 6.26

Table 4. Radiative habitable zones at Apoastron. Columns are: (1) HIP name, (2) Alternative name (some cases), (3) rHZ case described in section 2.1.2, (4)Inner rHZ for primary star, (5) Outer rHZ for primary star, (6) Inner rHZ for secondary star, (7) Outer rHZ for secondary star, (8) Inner rHZ for binary zoneand (9) Outer rHZ for binary zone. Note: rHZ - Radiative Habitable Zone.c©—- RAS, MNRAS 000, 1–??


HIP Alter. name a e m∗1 m∗2 r∗1ce r∗2ce r∗∗ rshift ref[AU] M� M� [AU] [AU] [AU] [AU]

– δ Equ 4.73 0.42 1.66 1.59 0.66 0.64 15.18 -0.07 A– V821 Cas 0.044 0.13 2.046 1.626 0.01 0.01 0.12 0.00 B

1349 HD 1273 1.25 0.57 0.98 0.55 0.13 0.1 4.18 -0.29 C1955 HD 2070 0.54 0.33 1.13 0.48 0.1 0.07 1.66 -0.11 C2941 ADS520 9.57 0.22 0.7 0.7 1.87 1.87 28.23 0.00 A5842 HD 7693 23.4 0.04 0.89 0.84 5.96 5.81 57.89 -0.07 D7078 HD 9021 0.64 0.31 1.21 0.7 0.12 0.09 1.97 -0.09 C7751 HD 10360 52.2 0.53 0.77 0.75 5.61 5.54 173.15 -0.54 D7918 HD 10307 7.1 0.42 0.8 0.14 1.26 0.57 22.13 -2.75 E8903 HD 11636 0.66 0.9 2.07 1.28 0.01 0.01 2.37 -0.18 E

11231 HD 15064 0.64 0.29 1.01 0.68 0.12 0.1 1.95 -0.06 C12062 HD 15862 2.04 0.26 0.95 0.44 0.43 0.3 6.11 -0.32 C12153 HD 16234 4.22 0.88 11 9.41 0.09 0.08 15.11 -0.39 E12623 HD 16739 1.27 0.66 1.39 1.13 0.1 0.09 4.35 -0.12 E14954 HD 19994 120 0.26 1.35 0.35 27.34 14.83 354.86 -28.27 F18512 HD 24916 174.55 0 0.35 0.17 52.36 37.68 315.65 0.00 G20087 HD 27176 7.05 0.17 1.76 0.95 1.66 1.26 20.08 -0.65 E20935 HD 28394 0.99 0.24 1.13 1.11 0.19 0.19 2.95 0.00 C24419 HD 34101 1.75 0.08 0.9 0.21 0.52 0.27 4.52 -0.17 C30920 4.3 0.37 0.22 0.08 0.773 0.488 13.421 -1.1144 K33451 HD 51825 9.3 0.43 1.61 1.26 1.31 1.17 29.93 -0.75 E34164 HD 53424 1.7 0.27 1.09 0.66 0.34 0.27 5.13 -0.19 C39893 1.81 0.21 0.95 0.52 0.4 0.31 5.29 -0.19 C44248 10.4 0.15 1.44 0.89 2.470 1.983 29.246 -0.6940 K45343 97.2 0.28 0.52 0.51 17.405 17.251 295.520 -0.4459 K56809 HD 101177 240.39 0.05 1.95 1.36 63.9 54.21 605.53 -5.06 G63406 HD 112914 1.59 0.33 0.82 0.23 0.32 0.18 4.86 -0.44 C64241 11.8 0.5 1.3 1.12 1.398 1.306 38.807 -0.6585 K64797 89.2 0.12 0.73 0.52 21.553 18.458 245.009 -3.5692 K67275 HD 120136 245 0.91 1.35 0.4 4.38 2.52 868.91 -147.85 F67422 32.7 0.45 0.72 0.65 4.306 4.109 105.959 -1.1531 K72848 HD 131511 0.52 0.51 0.93 0.45 0.07 0.05 1.71 -0.13 D73440 HD 133621 1.25 0.22 1.03 0.15 0.32 0.14 3.56 -0.30 C75379 HD 137502 0.91 0.68 1.26 0.68 0.07 0.05 3.12 -0.26 C76852 HD 140159 12.4 0.15 2 1.98 2.7 2.69 34.96 -0.02 E79101 HD 145389 2.24 0.47 3.47 1.31 0.33 0.21 7.23 -0.68 C80346 2.07 0.67 0.5 0.13 0.18 0.1 6.98 -1.04 C80686 HD 147584 0.12 0.06 1.05 0.37 0.04 0.02 0.3 -0.01 C82817 HD 152771 1.38 0.05 0.56 0.33 0.38 0.3 3.47 -0.04 E82860 HD 153597 0.33 0.21 1.18 0.52 0.08 0.05 0.96 -0.05 C84425 7.7 0.49 1.23 0.86 0.970 0.824 25.221 -0.9998 K84720 HD 156274 91.65 0.78 0.79 0.47 4.33 3.41 321.18 -24.55 D84949 HD 157482 4.87 0.67 2.62 1.15 0.29 0.42 16.6 -1.73 E86400 HD 1360346 0.39 0.23 0.72 0.39 0.08 0.06 1.15 -0.05 C86722 HD 161198 3.97 0.94 0.94 0.34 0.04 0.03 14.21 -2.18 E87895 HD 163840 2.14 0.41 0.99 0.68 0.32 0.27 6.84 -0.25 C91768 HD 173739 49.51 0.53 0.39 0.34 5.43 5.1 164.2 -2.67 G93017 ADS 11871 22.96 0.25 1.65 1.58 4.34 4.26 68.77 -0.21 A93825 32.7 0.32 1.27 1.25 5.463 5.424 101.163 -0.1364 K95028 HD 181602 0.85 0.37 1.4 0.5 0.15 0.1 2.65 -0.22 C95575 HD 183255 0.62 0.15 0.78 0.38 0.15 0.11 1.74 -0.06 C98001 HD 188753 11.65 0.47 1.3 1.11 1.48 1.38 37.98 -0.66 E99965 HD 193216 1.24 0.08 0.88 0.56 0.32 0.26 3.26 -0.05 C109176 HD 210027 0.12 0 1.25 0.8 0.03 0.03 0.22 0.00 C111170 HD 213429 1.74 0.38 1.08 0.7 0.28 0.23 5.5 -0.22 C113718 HD 217580 1.16 0.54 0.76 0.18 0.15 0.08 3.78 -0.51 C116310 HD 221673 95 0.322 2 2 15.7 15.7 294.14 0.00 H116727 HD 222404 18.5 0.36 1.59 0.4 3.55 1.9 57.04 -5.72 F

Kepler 16 0.22 0.16 0.69 0.2 0.06 0.03 0.63 -0.03 IKepler 34 0.23 0.52 1.05 1.02 0.03 0.03 0.76 -0.003 JKepler 35 0.18 0.14 0.81 0.81 0.040 0.040 0.504 0.0000 LKepler 38 0.15 0.1 0.95 0.25 0.043 0.024 0.397 -0.0163 LKepler 47 0.08 0.02 1.043 0.362 0.025 0.015 0.185 -0.0021 LKepler 64 0.17 0.21 1.528 0.408 0.042 0.023 0.490 -0.0333 L

Table 5. Orbital parameters and stability radii. Columns are: (1) HIP name, (2) Alternative name (some cases), (3) Semi-major axis, (4) eccentricity, (5)Mass of the primary star, (6) Mass of the secondary star, (7) Outer stable circumstellar radius for primary star, (8) Outer stable radius for secondary star, (9)Inner stable radius for circumbinary zone and (10) References about the orbital parameters. References are: A=Holman & Wiegert (1999), B=Cakirli et al.(2009), C=Jancart et al. (2005), D=Bonavita & Desidera (2007), E=Martin et al. (1998), F=Desidera & Barbieri (2007), G=Strigachev & Lampens (2004),H=Muterspaugh et al. (2010), I=Doyle et al. (2011), J=Lines et al. (2011), K=Haghighipour & Kaltenegger (2013) and L=Eggl et al. (2013)

c©—- RAS, MNRAS 000, 1–??


HIP Alter. Name case R∗1inner R∗1

outer R∗2inner R∗2

outer R∗∗inner R∗∗

outer

AU AU AU AU AU AU

– V821 Cas 3 - - - - 5.04 8.551955 HD 2070 3 - - - - 1.77 2.202941 ADS520 1 0.55 0.95 0.55 0.95 - -5842 HD 7693 1 0.87 1.5 0.78 1.34 - -7078 HD 9021 3 - - - - 2.06 2.677751 HD 10360 1 0.66 1.13 0.63 1.08 - -7918 HD 10307 1 0.71 1.22 0.03 0.04 - -8903 HD 11636 3 - - - - 5.01 7.66

12153 HD 16234 3 - - - - 128.82 219.8514954 HD 19994 1 1.92 3.3 0.15 0.25 - -18512 HD 24916 1 0.15 0.25 0.04 0.06 - -30920 1 0.06 0.1 0.01 0.02 - -44248 1 2.19 2.47 0.9 1.55 - -45343 1 0.31 0.54 0.3 0.52 - -56809 HD 101177 1 3.86 6.63 1.95 3.34 - -64797 1 0.6 1.03 0.31 0.54 - -67422 1 0.58 1 0.48 0.82 - -67275 HD 120136 1 1.92 3.3 0.19 0.33 - -80346 1 - - 0.03 0.04 - -80686 HD 147584 3 - - - - 1.23 1.9082817 HD 152771 1 0.36 0.38 0.14 0.24 - -82860 HD 153597 3 - - - - 1.63 2.4284720 HD 156274 1 0.69 1.19 0.26 0.44 - -91768 HD 173739 1 0.18 0.31 0.14 0.24 - -93017 ADS 11871 1 2.86 4.34 2.64 4.26 - -93825 1 1.72 2.94 1.66 2.85 - -95028 HD 181602 3 - - - - 2.87 3.25109176 HD 210027 3 - - - - 1.83 2.99116310 HD 221673 1 4.06 6.97 4.06 6.97 - -116727 HD 222404 1 2.62 4.5 0.2 0.32 - -

Kepler 16 3 - - - - 0.66 0.85Kepler 34 3 - - - - 1.67 2.81Kepler 35 3 . - - - 2.44 2.75Kepler 38 3 . - - - 0.71 0.97Kepler 47 3 . - - - 0.185 0.19Kepler 64 3 . - - - 2.6 4.41

Table 6. Binary systems with efective habitable zone. In these cases we can find radiative safe zones (at apo and periastron) within the stable zone. Columnsare: (1) HIP name, (2) Alternative name (some cases), (3) Case of HZ introduced in section 2.1.2, (4) Maximum of inner radii at Apoastron and Periastron forthe primary star intersected with the corresponding stability zone, (5) Minimum of outer radii at Apoastron and Periastron for the primary star intersected withthe corresponding stability zone, (6) Maximum of inner radii at Apoastron and Periastron for the secondary star intersected with the corresponding stabilityzone, (7) Minimum of outer radii at Apoastron and Periastron for the secondary star intersected with the corresponding stability zone, (8) Maximum of innerradii at Apoastron and Periastron for the circumbinary zone intersected with the corresponding stability zone, (9) Minimum of outer radii at Apoastron andPeriastron for the circumbinary zone intersected with the corresponding stability zone. Corrections because of the shift were taken in to account.

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Habitable Zones with Stable Orbits for Planets around